Arithmetices principia: nova methodo

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Mc^-th i?g-.tf

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SCIENCE CENTER LIBRARY

IL

FEOM THB PtJND OP

CHARLES MINOT

CLASS OF 1828

-^

iraims pmipii

//

NOVA METHODO EXPOSITA

lOSEPH PEANO

Izi X%. .i^oad.ezxa.ia miHtarl profesBore
.Aj3.alyBi33. in f1 -n itorvim in X%. Ta-uxiza.exajii ^tHezieeo dooezite.

AUGUSTAE TAURINORUM
Ediderunt FRATRES BOCCA

x^so-zs Bzsr.zos>ox.>^s

BOMAE PLOBENTJAE

Vift del Oorso, 216-2 17. V i a e r r e t & n i, 8.

1889

hiiut^ fuM.(i

lURIBUS SERVATIS

Augustae Taurinorum — Typis Vincentii Bona.

PRAEFATIO

Quaestioaes, quae ad matbematicae fundamenta pertinent, etsi
hisce temporibus a multis tractatae, satisfacienti solutione et adbuc
carent. Hic difficultas maxime en sermonis ambiguitate oritur.

Quare summi interest verba ipsa, quibus utimur attente per-
pendere. Hoc examen mihi proposui, atque mei studii resultatus,
et arithmeticae applicationes in hoc scripto expono.

Ideas omnes quae in arithmeticae principiis occurrunt, signis
indicavi, ita ut quaelibet propositio his taotum signis enuncietur.

Signa aut ad logicam pertinent, aut proprie ad arithmeticam.
Logicae signa quae hic occurrunt, sunt numero ad decem, quamvis
non omnia necessaria. Horum signorum usus et proprietates non-
nuUae in priore parte commum sermone explicantur. Ipsorum
theoriam fusius hic exponere nolui. Arithmeticae signa, ubi oc-
currunt, explicantur.

His notationibus quaelibet propositio formam assumit atque
praecisionem, qua in algebra aequationes gaudent, et a proposi-
tionibus ita scriptis aliae deducuntur, idque processis qui aeqaa-
tioaiim resolutiotti aasimilantur. Hoc capnt totius scripti.

Sique, confectis signis quibus arithmeticae propositiones scribere
possim, in earum tractatione lisus sum methodo, quam quia et
in aliis studiis tBoqumda foret, breviter exponam.

Ex arithmeticae signis quae caeteris, una cum logicae signis
exprimere licet, ideas significant quas deflnire possumus. Ita
amnia definivi signa, si quatuor excipias, quae in explicationibns
§ 1 continentur* Si^ ut puto, haec ulterius reduci nequeunt» ideas
ipsisexpressas^ideis quae priusnotae stipponiiatur, deflnirenon licet.

Propositiones, quae logicae operationibus a caeteris deducuntur,
Bunt theoremata; quae vero non, axiomata vocavi. Axiomata
hic sunt novecn (§ 1), et signorum, quae definitione carent, pro-
prietates fundamentales exprimunt.

In § 1-6 numerorum proprietates communes demonstravi; bre-
vitatis causa, demonstrationes praecedentibus similes omisi; de-
monstrationum communem formam immutare oportet ut logicae
signis exprimantur; haec transformatio interdum difficilior est,
tamen inde demonstrationis natura clarissime patet.

In sequentibus § varia tractavi, ut huius methodi potentia magis
videatur.

In § 7 nonnulla theoremata, quae ad numerorum theoriam
pertinent, continentur. In § 8 et 9 rationalium et irrationalium
definitiones inveniuntur.

Denique, in § 10, theoremata exposui nonnulla, quae nova
esse puto , ad entium theoriam pertinentia, quae cl.""' Cantor
Punktmenge (ensemble de points) vocavit.

In hoc scripto aliorum studiis usus sum. Logicae notationes et
propositiones quae in num. II, III et IV continentur, si nonnullas
excipias, ad multorum opera, inter quae Boolb praecipue, refe-
renda sunt (*).

(*) Boole: The mithematicaH anaJysis of logic, etc. Cambridge, 1847.

— The cdlculus of logic, Gamb. and Dablin Math. Joamal, 1848.

— An inveaiigaiian of the laws of thoughi, etc. London, 1854.
E. SchrOder: Der Operationskreia des Logikkdlculs, Leipzig, 1877.

Ipse iam nonnalla qaae ad logicam pertinent tractavit in praecedenti opera.

— Lehrhuch der Arithmeiik und Algebra, etc. Leipzig, 1878.

Boole e SchrOder tbeorias brevissime ezposai in meo libro Caholo geometrieo etc.
Torino, 1888.

Vide:

C. S. Peirce, On the Algehra of logic; American Joumal, III, 15; VII, 180.

Jevons. The principlea of science, London, 1885.

Mc.CoLL. The edlculus of equivalent statements. Proceedings of the London
Math. Society, 1878. Vol. IX, 9. Vol X, 16.

Signum €, quod cam sigoo q confandere non licet, inversionis
in logica applicationes, et paucas alias institui conventibnes, ut
ad exprimendam quamlibet propositionem pervenirem. .

In arithmeticae demonstrationibus usus sum libro: H. Grass-
MANN, Lehrbuch der Arithmetik, Berlin 1861.

Utilius quoque niibi fuit recens scriptum: R. Dedekjno, Was
sind und was Mollen die Zahlen; Braunscbweig, 1888, in quo
quaestiones, quae ad numerorum fandamenta pertineot, acute exa-
minantur.

Hic meus libellos ut novae methodi specimen habendus est.
Hisce notationibus innumeras alias propositiones, ut quae ad ra-
tionales et irrationales pertinenty enunciare et demonstrare pos-
sumus. Sed, ut aliae theoriae tractentur, nova signa, quae nova
indicant entia, instituere necesse est. Puto vero his tantum lo-
gicae signis propositiones cuiuslibet scientiae exprimi posse, dum-
modo adiungantur signa quae entia huius scientiae representant.

SiaNOBnU TABULA

Sigpiuin
P

K
n
u

A'

€

[]
3

Th
Hp
Ts
L

LOGIGAI SiaNA

Signifieatio
propositio
cktssis
et
vel
non

Pag-

VII
X

vn, X
vm, X, XI

VIII, X

absurdum aut nihil viii, xi
dedncitur aut confmefur viii, xi

esi MqucUis viii

est X

inversionis signum xi

qui vel [e] xn

Theorema xvi

Hypothesis »

Thesis »

Logica »

ARTniMirriCAE sigka

Signa 1, 2, ..., =, >, <, +, -, X

vulgarem habent significationem. Di-
visionis signum est /.

Signttm
N
R
Q

Np
M

W
T
D

a

SlGNA GOMPOSITA

Significaiio Pag.

numerus integer positivus 1

num. rationalis positivus 12
quantitas, sive numerusrea-

lis positivus 16

numerus primus 9

maosimus 6

minimus 6

terminus, vel limes summus 15

dividit 9

est multiplex 9

est primus cum 9

— < non est minor

= u > 65^ aequalis aut maior

3 D divisor

M^D moMimus divisor

Logicae notationes.

I. De punctuatione,

Lltteris a, &,... x, «/,... x*y\„ entia indicamus indeterminata quae-
cumque. Entia vero determinata signis, sive litteris P, K, N,... in-
dicamus.

Signa pleruraque in eadem linea scribemus. Ut ordo pateat
quo ea coniungere oporteat, parenlhesibus ut in algebra, sive
punctis . : .*. : : etc. utimur.

Ut forraula punctis divisa, intelligatur, priraura signa quae nuUo
puncto separantur colligenda sunt, postea quae uno puncto, deinde
quae duobus punctis, etc.

Ex. g. sint a, &, c,... signa quaecumque. Tunc ab.cd significat
{ab){cd); et ab.cdief.gh .-. h signiflcat {{{aX>){cd)){{ef){gh)))n.

Punctuationis signa oraittere licet si formulae quae diversa pun-
ctuatione existerent etindera habeant sensum; vel si una tantum
forraula, et ipsa quara scribere voluraus, sensum habeat.

Ut ambiguitatis periculum absit, aritmeticae operationum signis . :
nunquam utimur.

Parenthesum figura una est ( ); si in eadem forraula, parentheses
et puncta occurrant, priraura quae parenthesibus continentur, col-
ligantur.

II. De prapositionibus.

Signo P significatur propositio.

Signura n legitur et. Sint a, b, propositiones ; tunc anb esi si-
multanea aflJrmatio propositionura a , b. Brevitatis causa, loco a n b
vulgo scriberaus ab.

— VIII —

Signum — legitur won. Sit a quaedam P; tunc - a est negatio pro-
positionis a,

Signum u legitur mL Sinta, & propositiones; tunc auft idem est
ac — : — a . — &.

[Signo V signiflcatur verum, sive identUas; sed hoc signo nun-
quam utimur].

Signum a significat falsum, sive absurdum,

[Signum G signiflcat est consequentta; ita & G a legitur b est con-
sequentia propositionis a, Sed hoc signo nunquam utimur].

Signum o signiflcat deducitur; ita a o & idem signiflcat quod
& G a. Si propositiones a, 6 entia indeterminata continent a?, j/,..., sci-
licet sunt inter ipsa entia conditiones, tunc aoo;, y,... 2^ significat:
quaecumque sunt x, y,..., a propositione a deducitur &. Si vero am-
biguitatis periculum absit, loco o»,y.... scribemus solum o-

Signum = signiflcat est ojequaXis, Sint a, & propositiones; tunc
a = & idem signiflcat quod a o &• & D ^/ propositio a = «?, y,...& idem
signiflcat quod a o «. y.... &. & 0«, y,... a.

III. Logicae propositiones.
Sint a, &, c,... propositiones. Tunc erit :

1.

ao<3J.

2.

ao&-&Oc:o:«OC.

3.

a = &. = :ao&-&0«-

4.

a = a.

5.

a = b, = ,b = a.

6.

a = b.J)oc:o,aoc.

7.

aob,b = c:0'^DC,

8.

a = b .b = c:o ,a = c.

9.

a = & . . a &.

10.

a = & . . & «.

11.

a^O^.

12.

a& = ba.

13.

a {bc) = {ab) c = abc.

— IX

14.
15.
16.
17.
18.
19.

20.
21.
22.

23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

a=:b .0'^ = ^'
a 6 . . ac &c.
a fe . c 6? : . ac M.
^ D & • ^ D c : = . a ()C.
a = & . c = t? : . ac = M.

-(-a) = a.
ao&. = .-&o-a.

au&. = .-. — : — a. — &.

-(a&) = (-a)u (-/>).

-(au&) = (-a)(-?)).

a . a u &.

a u & = & u a.

a u (& u c) = (a u ?/) u c = a u ?? u c.

a u a = a.

a (& u c) = a& u ac.

a = i5?.o.auc = &uc.

ao^.O.auco&uc.

ao&-^0^:D-«^c-0-&'-'^-

&Oa-co«: = -&'-'^0«-

35.

a-a = A.

36.

«A = A-

37.

auA. = <*'

38.

« D A = • a = A-

39.

ao6. = .a-d = A-

40.

AD«-

41.

au6 = A- = :« = A-& = A'

42. a • & c : = : ai5? <^-

43. a . ^ = c : = . a6 = oc.

— X —

Sit a quoddam relationis signum (ex. gr. =, q), ita ut a a & sit
quaedam propositio. Tunc loco — . a a & scribemus a-^ab; soilicet:
a — = 6. = : — .« = &.
a-06. = :-.ao&.
Ita signum - = signiflcat non est oeqtMUis. Si propositio a inde-
terminatum continet a?, a — =rA sJgnificat: sunt o? quae conditioni
a satisfaciunt. Signum — o significat non deducitur.

Similiter, si a et p sunt relationis signa, loco aab.a^b, et aa
& . vj . a 3 & scribere possumus a.a3.6eta.awp.&. Ita, si a et fe
sunt propositiones, formula a . o — = . ^ dicit: ab a deducitur &, sed
non vice versa.

a.Q — = .&: = :ao&.& — oa.
Formulae:

ao&-&OC.a-oc: = A.-
a = h .h = c.a '- = €: = h^.
ao2?.2?0- = c:o-«D- = c.
ao- = &.^Oc:o.«0- = c.
Sed his notationibus raro utimur.

IV. De cUissibus.

Signo K significatur classiSy sive entium aggreg^tio.

Signum e significat est. Ita a e & legitur a est quoddam b ; aeK
significat a est quaedam classis; aeP significata est quaeda^n
propositio.

Loco — (a € &) scribemus a — e 6 ; signum — e significat non est ;
scilicet :

44. a^eb. = :^.aeb.

Signum a,b,cem significat : a, b et c sunt m ; scilicet :

45. a,b,cem. = :aem. bem.cem.

Sit a classis; tunc — a significatur classis individuis constituta
quae non sunt a.

46. a € K . : i» e — a . = . 0? - e a.

Sint a, b classes ; a n &, sive a b, est classis individuis constituta

quae eodem tempore sxxnX a et &; a u & est claasis individaiis con-
stituta qui sunt a vel b.

47. a, & € K . /. 0? 6 . a & : = : ^ 6 a . 0? c &.

48. a, & 6 K . 3 .•. a? € . a u & : = : 0? € a . u . a? € &.

Signum ^ indicat classem quae nuUum continet individuum. Ita :

49. a 6 K . .'. a = A • = • ^ e a . =a> A-

[Signo Y, quod classem ex omnibus individuis constitutam, de
quibus quaestio est» indieaity non utimur].

Signum ^ signiflcat contineiur. Ita a o & signiflscat classis a con'
tinetur in classi &.

50. a, & 6 K . .*. a Q & : = : o? € a . 0« . a? € &.

[Formula h(\a signiflcare potest classis b continet classem a;
at signo G non utimurj.

Hlc signa A et o signiflcationem babent quae paulio a praecedenti
difiG^; sed nulla orietur amfaiguita& Nam si de proposiftiombus
agatur, haec agna legantur o^bsurdMm et dedMCituv; » vero de
classibus, nUiU et contineLur.

Formula a = &, si a et & sint classes, significat a o 6 . & o a. Itaque

51. a, & € K . .'. « = ^ ^ = ^ a? € a . =0; .x^h,

Propositiones 1... 41 quoque subsistunt, si a, &... classes indicant;
praeterea est:

52. ae&.o-^cK:-

53. a € & . o . 2> - = A-

54. a e & . & = c : • <3J € c.

55. ac&. &o<?^D.«ec.

Sit s classis, et K classis quae in s contineatur; tunc dicimus.ft
esse individuum classis s, si k ex uno tantum constat individuo.
Itaque:

56. 5 € K . /t o^ ^ D "* ft e 5 . = .'. fe — = A • ^» 1/ € fc . o«,y . a? = !/•

V. De inversione.

Inversionis signum est [], eiusque usum in sequenti numero ex-
plicabimus. Hic tantum casus particulares exponimus.

— XII —

1. Sit a propositio, indetermlnatura continens x\ tunc scriptura
[a?] € a, quae legitur ea x quihus a, sive solutiones, vel radices con-
ditionisa, classem significat individuis constitutam, quae conditioni a
satis&ciunt. Itaque:

57. aeP.o:[a?e]a.€K.

58. a 6 K . Q .-. [a? e] . a? € a : = a.

59. a € P . .-. a? € . [a? e] a : = a.

Sint a, p, propositiones indeterminatum continentes x; erit:

60. [xe]{a&) = {[xe]a){[xe]?).

61. [a?€]-a = -[a?6]a.

62. [xe] (a up) = [xe]asj[xe] p.

63. a 0» p . = . [a? €] a [a? c] p.

64. a=x?> = '[xe]a = [x e] p.

2. Sint X, y entia quaecumque; systema ex ente x et ex ente y
compositum ut novum ens consideramus, et signo (a?, y) indicamus;
similiterque si entium numerus maior fit. Sit a propositio indeter-
minata continens x,y; tunc [{x, y) e\ asigniflcatclassem entibus (Xyy)
constitutam, quae conditioni a satis&ciunt. Erit:

65. a oa,, y p . = . [{X, 2/) €] a [{x, y)e]^.

^' [{ccy t^) €j a - == A . = .-. [a? €] . [y €] a - = A : - = A-

3. Sit xay relatio inter indeterminata x eX y (ex. g. in logica
relationes a? = 2/, a? — = y, a? o y ; in arithmetica x <y,x>y, etc).
Tunc signo [e a] ^ ea ^ indicamus, quae relationi xay satisfaciunt.
Gommoditatis causa , loco [e], signo 9 utimur. Ita 9 a y . = : [^ e] .
xay, et signum 3 iegitur qui, vel quae. Ex. gr. sit y numerus;
tunc 9 < 1/ classem indicat numeris x compositam qui conditioni
X <y satisfaciunt, scilicet, qui sunt minores y, vel simpliciter
minores y. Similiter, quum signum D signiflcet dUvidit, vel esi di-
visor, formula aD signiflcat qui dividunt vel divisores. Deducitur
xe3ay = xay.

4. Sit a formula indeterminatum continens x. Tunc scriptura
a?' [a?] a, quae legitur a?' loco x in a subsiituto, formulam indicat
quae obtinetur si in a, loco x, ai legimus. Deducitur x[x]a = a.

5. Sit a formula, quae indeterminata x, y^... continet. Tunc

(a?', t/',...) [x, 1/,...] a.

— XIII —

quae legitur al i/,... loco x, y,... in a substttutis, formulam indicat
quae obtinetur si in a loco a?, y,..., litterae a?' y',... scribantur. Dedu-
citur (a?, y)[x,y]a = a.

VI. Z)e /wnc/^one^&t^^.

Logicae notationes quae praecedunt exprimendae cuiiibet arithme-
ticae propositioni sufflciunt, iisdemque tantum utimur. Hic nota-
tiones alias nonnullas breviter explicamus, quae utiles fieri possunt

Sit s quaedam classis ; supponimus aequalitatem inter entia syste-
matfl s deflnitam, quae conditionibus satisfaciat:

a = b. = .b = a.

a = b.b = c:^,a = c.
Sit qp signum, sive signorum aggregatus, ita ut si x est ens
classis Sj scriptura qpa? novum indicet ens; supponimus quoque
aequalitatem inter entia qp x deSnitam ; et ^i x eX y sunt entia
classis 5, et est x = y, supponimus deduci posse q>x = q>y. Tunc
signum q> dicitur esse functionis yraesignum in classi 5, et scri-
bemus (paF*^.

S€K.o''(p^^*s. = .\Xyyes.x = y:ox,y*(px = (py.
Verum si, cum sit x quodlibet ens classis 5, scriptura X(p novum
indicet ens, et, exx = y deducitur X(p = ycp, tunc dicimus cp esse
functionis postsignum in classi s et scribemus cp e s'F.

5 6 K . :: cp € 5 'F . = .-. o?, j/ € 5 . 07 = 2/ : Qo;. y . a?q) = i/(p.

Exempla. Sit a numerus; tunc a+ est functionis praesignum in

numerorum classe, et + a est functionis postsignum ; quicumque enim

est numerus x, formulae a + x et a? + a novos indicant numeros,

et ex x = y deducitur a-^-x^a-^-y^ eta? + a = y + a. Itaque

a6N.o:a + .€.F*N.

a € N . : + a . e . N 'F.

Sit qp functionis praesignum in classe s. Tunc [qp] y classem signi-
ficat iis X constitutam, quae conditioni <^x = y satisfaciunt ; scilicet :
Def. 5 € K . qp 6 F* 5 : D : [qp] y . = . [a? e] (qp a? = y).

XIV —

Glassis [cp] y vel unam Tel plura, vel ettam nuUum indrvidanm
contineFe potest. Erit:

5 € K . cp 6 F* 5 : : 2/ = q) a? . = . a? € [(p] j/.
Si vero cp y uno tantum constat individuo, erit y = cp it? . = . ^

= W\ y-

Sit (p functionis postsignum; similiter ponimus:

5 € K . <p € 5 'F : . • . t/ 1 cp I = I ^ € I («? <P = y )•

Signum [] dicitur (nversionis sigmum, eiusque usus nonnullos
in logica iam exposuimus. Nam si a est propositio indeterminatQm
continens x, atque a est classis individuis a? compostta quae con-
ditioni a satisfaciunt, erit a?€a. = a, tunc a = [a?6]a, utin V, 1.

Sit a formula indeterminatum continens x, sitque cp functionis
praesignum, quod iitterae a? praepositum, formulam a gignat; sci-
licet sit a = q>x; tunc erit (p = a[x], et si zp' est novum ens, erit
<p 0?' = a [x] x\ scilicet, si a est formula indeterminatum continene x^
tunc a \x] af significat id quod obtinetur si in a, loco a?, a?' po-
natur.

Similiter, sit a formula indeterminatum continens x, sitque a^ fun-
ctionis postsignum, ut a? <p = a; deducitur cp ;= \x\ a ; tunc, si af est
novum ens, erit x' cp = a/ [x] a, scilicet af [x] a rursum indicat id
quod obtinetur «i in a, loco a?, a/ legatur, ut in V, 4.

Alios quoque usus in logica signum f ] liabere potest, quos bre-
viter esponimus, quum ipsisnon utamur. Sint a et & duae classes;
tunc [a n ]i) sive b[ n a] classes indicat x, quae conditioni b=anx,
sive b = xna satisfaciunt. Si & In a non continetur, nulla classis
huic condltionl satisfacit; si ?^ in a oontinetur, signum b[ n a\ omnes
indicat classes quae b oontinent atque in &u — a continentur.

(n A.rlthraetlca, sint a, b numeri ; tunc [b +^] sive [a +] b nu-
merum indicat x, qui conditioni b = x-]-a, sive b = a-f-x sa-
tisfacit, nempe b — a. Similiter erit b[y(^a] = [ay]b = bja. Et in
analysi hoc signum usuvenire potest; itaque

y = s\nx. = .xe [sin] y (loco x = arc sin y)

dF (x) = f{x) dx. = .F{x)E[d\ f{x) dx (loco F {x) = ff{x) dx).

Sit rursum qp functionis praesignum in classi ^, sitque h classis

in s ccmt&Qta; tunc <p h clasaem indicat omnibus <|> (v compositam, ubi
X sunt entia classis ft; scilicet

Def. .9 € K . ft e K . ft 5 . cp e F* 5 : . <p A = [i/ ej (ft . f qpj ?/: - = a).
Sive seK.heK.kos .q>eF's:o.(phsa [ye] ([A?e] :xeh.q>iv

=y '''- = A)'
Def. S€K.feeK.fto/?.<p€5'F:o.A(p = [|/ej(ft.|/[cp]:-.=:A)-
Itaque, si (p € F' ^, tune (p s classem indicat omnilms (p x constt-
tutam, ubi x sint entia classis s. Erit:

5 6 K . <p e F* ^ . y € (p 5 : : (p [(p] y = y.

5€K.a,&€K.a3 5.ftO^.(p€F*5:3.<p(avj&)=(<pa)u(^&).

5€K.(peF*5:o.(pA = A.
^•eK.a, &eK.&o*-«D&.<P€F*5:o-<P<*D9&-
5€K.a, &€K.ao5.&D^-q>6F'5:o.<P (^) D (q> «) (q> V).
Sit a quaedam classis; tunc anK, sive Kna, sive K«, classes
omnes indicat formae anx, sive xna, xa, ubi a? est classis quae-
cumque; scilicet K a indicat classes quae in a contineatur. Formula
a?€Ka idem significat quod xeK.x^^a. Hac conventione quan-
doque utimur; ita KN significat numerorum classem,
Similiter, si a est classis, K u a indicat classes quae a continent.
Sit a numerus ; taiHc a + N, sive N + <^» numeros indicat mmero
a maiores; aXN, sive NX«> siv® N^i indicat multtpUces nvr
mjeri a\ a^ indicat potestaies numeria; N*, N^... indicant numeros
quadratos, vel num^ros oubos, etc.

Fun<^ionum signorum aequalitatem, productum, potestates, ita
definire licet:

Def. .9 e K . (p, ip € F' ,s : «•• 9 = ^* : = • ^ € 5 . • <P ^ = M^ ^.
Def. 5€ K . (p e F*^ . q) e F* (p 5 . fl? e 5 : o • H' cp a? = ip (<p a?).

Itaque, in definitionis hypothesi, erit vp cp novum functionis prae*
signum; idque producttmi signorum qi e^ cp vocatur.

Similiter<|ue, si <p, tp sunt functionis postsigna.

Haec valet propositio:

5€K.(p>eF*5.(p^05^0'<P950^.q??950*. e*c.
Functiones qp qp, (p (p (p,... iteraiae vocantur, et communiter signis
cp*, cp^... indicantur, ut operationis cp potestates.

— XVI —

Si vero qp est functionis postsignura, hac faciliori notatione, absque
arabiguitate, uti licet:
Def. 5€K.q)€5'F.5q)35:o:(pl = (p.q)2 = (p(p.q)3=: q)q)q).etc.

In deflnitionis hypothesi, si m, n € N, erit 9 (m-\-n) =r (q) w) (q)n) ;
{(^m)n = fp{m n).

Si hac definitione in Arithraetica utimur, haec invenimus. Nu-
raerura qui sequitur nuraerum a signo faciliori a+ indicare pos-
suraus; tunc a + ^j^ + ^,... et, si b est numerus, a + &, sonsum
habent a +, a + +,... quod a deflnltione in § 1 patet. Propositionera 6
in § 1 scribere possumus N + N. Si a, 6, c sunt numeri, tunc
a:-\-b ,c significat a + &c, et a:X^-c significat a b<^.

Multis aiiis proprietatibus gaudent functionum signa, praesertim
si conditioni satisfaciunt : q) x ='q) y .Q,x = y. Functionis signum
quod huic conditioni satisfacit vocatur a clarissimo Dedekind sfmile
(ahniich Abbildung).

Sed his exponendis locus deest.

Declarationes.

Deflnitio, vel breviter Def, est propositio formam habensa? = a,
sive a 3 . a?=a, ubi a est signorum aggregatus sensum habens notum ;
X est signum, vel signorum aggregatus significatione adhuc carens;
a vero est conditio sub qua definitio datur.

Theorenia^ (Theor. vel Th) est propositio quae demonstratur. Si
theorema forraam habet aop, ubi a et p sunt propositiones, tunc
a dicitur Hypothesis (Hyp. vel breviter Hp.), p vero Thesis (Thes.
vel Ts.). Hyp. ac Ts. a Theorematis forma pendent; nara si loco
aoP scriberaus— Po— a, erit— pHp,et — aTs.; si vero scribemus
a — 3= At Hp. ac Ts. absunt.

In quolibet § signura P quod quidam numerus sequatur, propo-
sitionera indicat eiusdera § hoc numero signatam. Logicae proposi-
tiones indicantur signo L et propositionis numero.

Formulae quae in una linea non continentur, in altera linea, nullo
interposito signo, sequuntur.

ARITHMETICES PRINCIPIA.

§ 1. De nnmeris et de additione.

E{vplicationes.

Signo N significatur numerus (integer positfvusj.
»1 » unitas.
» a-^i > sequms a, sive a plus 1.
» = » esi aequalis. Hoc ut novum signum conside-
randum est, etsi iogicae signi figuram habeat.

Aociomata.

1. l€N.

2. a € N . . a = a.

3. a, &, c € N . : a = & . = . & = a.

4. a, & € N . .-. a = & . & = c : . a = c.

5. a = &.&6N:o.aeN.

6. aeN.o.a + lcN.

7. a, & e N. : a = & . = . a + 1 = & + 1.

8. aeN.o.a + l- = i.

9. ft e K .-. 1 e /c .-. 0? e N . 07 e A: : Oa? . a? + 1 e ft : : . N A:.

10. 2 = 1 + 1; 3 = 2+1; 4 = 3 + 1; etc.

Pbamo, Arithmetiees principia. 1

leN

(1)

leN.Q.l + leN

(2)

l + l€N

(3)

2=1+1

(4)

— 2 —

Thecyremata,

11. 2eN.
Demonstratio:
Pl.o:
l[a](P6).D:
(i)(2).D:
PIO.d:
(4) . (3) . (2, 1 + 1) [a, b\ (P 5) : o: 2 e N (Theorema).

Nota, — Huius facillimae demonstrationis gradus omnes explicite
scripsimus. Brevitatis causa ipsam ita scribemus:

P 1 . 1 [a] (P6) : : 1 + 1 eN . P 10.(2,l + i)[a,&](P5):o:Th.
vel

Pl.P6:o:l + leN.P10.P5:o:Th.

12. 3, 4,... e N.

13. a, &, c, t;? e N . a ==&.& = c . c = rf : :« = ^.
Dem, Hyp. P 4 : o : «, c, rf e N. a = c . c = rf . P4 : o : Thes.

14. a, &, c 6 N . a = & . & = c . a — = c : = A-
Dem. P 4 . L 39 : 0. Theor.

15. a, &, c € N . a = & . & — = c : . « — = ^*.

16. a^&eN.a^&^o.^ + i^^^+i.
16'. a, & e N . a + 1 = & + 1 : . « = ^.
Dem. P7 = (P16)(P16').

17. a,&€N.o:«- = &. = .« + i- = ^ + i.
Dem. P 7 . L 21 : . Theor.

Definitio.

18. a,&€N.o.a + (& + l) = (a + &) + i.

Nota, — Hanc deflnitionem ita legere oportet: si a et & sunt
numeri, et (a + &) + 1 sensum habet (scilicet si a + & est numerus),
sed a+(& + l) nondum definitus est, tunc a + (& + !) signiflcat
numerum qui a + & sequitur.
Ab hac deflnitione, et a praecedentibus deducitur:
aeN.o.-.a + 2 = a + (l+l) = (a + l) + l.
a 6 N . .'. a + 3 = a + (2 + 1) = (a + 2) + 1, etc.

3 —

Theoremata,

19. a,&€N.D.a + &€N.

Dem. aeN.P6:o:a + l6N:o:l€[&e]Ts. (1)

a e N . :.•& 6 N . & € [& el Ts : : a + &6 N . P 6 :o : («+&) +

leN.P18:o:a + (& + l)eN:o:(2> + l)€[&e|Ts. (2)

aeN.(l).(2).o::l€[&€lTs.-.&eN.&€[&e]Ts:o:&+l€[&e]

Ts.-.([&e] Ts) [/jjP9::o:No[&€] Ts . (L50) ::o:&€N.o

Ts. (3)

(3) . (L 42) : : a, & € N . . Thesis. (Theor.).

20. i>g/: a + & + c =: (a + &) + c.

21. a, &, c € N . . « + 2? + c € N.

22. a, &, C€N.o:« = &. = -« + c = & + c.

Dem. a,&eN.P7:o.l€[c€]Ts. (1)

a, & € N . : : c € N . c € [c e] Ts .-. .•.a=& .=.a+c=& + c :

a + c, &+C€N:a+ c = 1>-\-g .=.a + c + l=&+c +

l.-.O.-.a = &.==.a + (c + l) = & + (c+l).-.0.-.(c + l)

e[ce]Ts. (2)

a, & € N . (1) .(2) : : : 1 € [c e] Ts.-. c € f c €] Ts .0 . (c + 1) € [c € 1

Ts::o::6'€N.o.Ts. (3)

(3) Theor.

23. a, &, c € N . . <3t + (& + c) = a + & + c.

Dem, a, & € N. P 18 . P 20 : . 1 e [c e] Ts. (1)

a, &€N.o.'. ceN.ce [ce] Ts:o:« + {p + c^^a-^-b+c.

P7:o:a + (& + c) + l = a + & + c + l.P18:o:a +

(& + (c + l)) = a + & + (c + l):o.c + le[c€]Ts. (2)

(l)(2)(P9).o.Theor.

24. a€N.o.l+«==« + l.

Dem. P 2 . . 1 € [a e] Ts. (1)

a€N.ae[a€]Ts:o:l+« = « + l:D:l + («+l)=(a +
l)+l:D:(« + l)€[ae]Ts. (2)

(l)(2).o.Theor.
24'. a,&€N.o.l + a + & = a + l+&.
Dem, Hyp. P24:o:l+a = a + l.P22:o. Thesis.

25.

— 4 ^

^. a, &€N.o.a + & = & + a.

2>em. a€N.P24:o:l€[&€]Ts. (1>

a € N . .'. & € N . & € [& €] Ts: : a +& = & + a . P 7 : :(a +
&) + l=(& + a) + l.(a + &) + l = a + (&+l).(& +
a)+l=l+(& + a).l + (& + a) = (l+&) + a.(l + &)
+ a = (&+l) + a:o:a+(& + l)=(& + l)+a:o:(&

+ 1)€[&€]TS. (2>

(l)(2).o.Theor.

a, &, C€N.o:« = &. = .c + a = c + &.

a,&,ceN.o:a + & + c = a + c + &.

a, &, c, deN.a = &.c = d:0.« + c = & + tf.

§ 2. De snbtractione.

26.

27.
28.

^a?p//catonc5.

Signum — legitur minus^
» < » esi minor.
» > » 65^ mxiior,

Definitiones.

1. a, & € N . : & — « = N [a? €] (07 + a = &).

2. a, fteN.o :«<&. = . 2> — « — = A.

3. a, & € N . : 2? > a . = . a < &.

a + h — c^^ia-^-b) —c\a — l) + c=^(a — li)-\-c\a — l) —
c = {a — V) — c.

Theoremata.

4. a, &, a', y e N . a = a' . & = &' : : 2? — a = y — a'.
i)e?n. Hyp .o:^ + a = &. = .^ + «' = &':D. Thesis.

5. a, &eN.o:<3^<&. = -& — «eN.

Bem. a, &€N:o.*.^, 2/€& — a.Oa;.y:o?, j/€N.a; + a=&.i/+a =

&.§lP22:o:^ = l^. (1)

a,fteN.a<&.P2.(l):o.'.& — «- = A:^»2/€& — a.o-^

= 2/ : (N, & - a) [5, /c] (L 56) .-. o •'. & — a € N. (2>

— 5 —

a, & € N . & — a € N . (L 56) : : & — a - = A : * « < &• (3)
(2)(3).o.Theor.

6. a, & 6 N . a < & : . & — a + « = ^-

Dem. Hyp . P 5 . P 1 : o : & — a e N . (& — a) € [o? ej (o? + a = &) : o :
Thes.

7. a,&, C€N.o:c = & — a. = .c + a = &.

Bem. Hyp . §1P 22. P6 : o* c = & — a. = .c + a = & — a+a.=
. c + a = &.

8. a, &€N.o.a + & — a = &.
2>em. (a + &,&)[&, c] P 7 . . Theor.

9. a, &, c € N . a < & : : c + (& — a) = c + & — a.

Dem. Hyp .P6:o^(&— a) + a = &:o-c + (& — a) + a = c + &.
P 7 : : Thesis.

10. a, &, c € N . a > & + c : • « — (& + c) = a — & — c.

11. a, &, C€N.&>c.a>& — c:o.<3J — (& — c) = a + c — &.

12. a, &, a', &' e N . a = a' .& = &': ••«< ^ . = •«' < V.

Dem, Hyp .o.&— -a = &'--a'.o.& — « €N=&' — a' €N.o .Thes,

13. a, & € N . . « < a + &•

i>em. Hyp .P8:o:a + & — a = &:o.<^ + ^ — «€N.P5:o:
Thesis.

14. a, &, c € N . a < & . & < c : • ^ < c.

i)em. Hyp . o •' & — a € N . c — & € N : o •' (& — «) + (c — &) € N : o : c
— a € N : . Thesis.

15. a, &, ceN.o:«<&. = .<a^ + c<& + c.

Dem. Hyp .o:a<&. = .& — a€N. = .(& + c) — (a + c)€N.=.
a + c < & + c.

16. a, &, a', &' € N . a < & . a' < &' : . a + «' < ^ + V.
Dem. Hyp .o:a + a'<& + a'.& + a'<& + &':o. Thesis.

17. a, &, c e N. a < & < c : . c — a > c — &.

Dem. Hyp .o.& — «€N.c--&€N.(c — &) + (& — a) = c — a:o.
Thesis.

18. a € N . : « = 1 . ^ • <3^ > 1-

i>cm. 1 e [a e] Thesis. (1)

aeN.P13:o:a + l>l:D:« + i€[ael Thesis. (2)

(l)(2).o.Theor.

— 6 —

19. a,&€N.o.a + &- = &.

Dem. a€N.§lP8:o:a + l- = l:D:le[&€] Thesis. (1)

a€N.&€N.&€[&€]Ts:D:a + &- = &.§! P17:o:a+(&
+ l)- = 6 + l:o:& + l€[&€jTs. (2)

(l)(2).o.Theor.

20. a, & € N . a < & . a = & : = A-

Dem. Hyp : o : & — <a^ 6 N . (& — a) + a = a . P 19 : o : A-

21. a, & € N . a > & . a = & : = A-

22. a,&€N.a>&.a<&: = A-

23. a, &€N:o:«<&. *-»•« = &. *-»•«>&•

Dem. a € N . P 18 : . 1 € [& €] Ts. (1)

a, & € N . a < & : . a < & + 1 • (2)

a,&€N.a = &:o. «<& + !. (3)

a, &€N.a>&:o:«-~&€N.P 18 :o:a — & = !.'-'•« — &
>!• (4)

a,&€N.a — & = l:o. « = & + !. (5)

a,&€N.a — &>l:o^«>& + l. (6)

a,&€N.a>&.(4)(5)(6):o:« = & + l.^^a>& + l. (7)
a,&€N:a<&.u.a = &.u.a>&:(2)(3)(7).-.o.'.a<& + l
.u.a = &+l.u.a>& + l. (8)

a, & € N . & € [& €] Ts . (8) : : & + i € [& €] Ts. (9)

(l)(9).o.Theor.

§ 3. De mazimis et minimis.

Eooplicationes.

Sit a € K N, hoc est sit a quaedam numerorum classis ; tunc Ma
legatur maodmus inter a, et pja legatur minimus inter a.

Deflnitiones.

1. a € K N . : M a = [a? € I (a? € a .-. a . 3 > 0? : = a)-

2. a e K N . . FI « = [a? € I (o? € a .-. a . 3 < a? : = a).

— 7 —

Theoremata,

3. neN.a€KN.a-=:A-^5>n = A-D-Ma€N.

Dem. acKN.a— = A-«5>l=A-0-<3^ = i-D-Ma = l:o.M

a e N. • (1)

(l)D:l€[ncl(HpDTs). (2)

neN.aeKN.a? >n + i = A-^ + l€a:o:n+l = Ma

: : M a € N. (3)

n€N.a€KN.a9>n + l = A-w + l — €a:o:a3>n

= A. (4)

n € [n ej (Hp oTs^.aeKN.aa^n+lr^A-^ + l-ca:

OiMaeN. (5)

n e [n e] (Hp Ts) . a e K N . a 9 > n + 1 = A . (3) (5) : : M a

eN. (6)

n^[n^] (HpoTs).(6) :o.(n + 1) e [ne] (HpoTs). (7)

(2) (7) . § 1 P 9 : : w € N . . Hp Ts. (Theor.)

4. a€KN.a- = A:0W«eN.

5. a € K N . .'Fi « = M [ic? e] (a 3 < a? = a).

§ 4. De mnltiplicatione.

DefiniUones.

1. a€N.o.«Xl = «.

2. a, & e N . . a X (2^ + 1) = « X 2^ + «.

a& = aX&;^ + c = (a&) + c ; dbc = {ab) c.

Theoremata.

3. a, & € N . . a& € N.

Z>em. a € N . P 1 : : « X 1 € N : . 1 € [& e] Ts. (1)

a,&cN.&€[&€]Ts:o:aX&6N.§lP19:o:«& + a€N.
Pl:o:a(«? + l)eN:o:& + l€[&6lTs. (2)

(l)(2).o.Theor.

— 8 —

4. a, &, c e N . . (a + &) c = ac 4" 2>c-

Nota. Haec est prop. 5* Euglidis elem. libri VII.

Dem, a,&6N.Pl:o:l€fc€]Ts. (1)

a, &, c € N . c € [c €] Ts : : (a + &) c = ac + &c . § 1 P 22 : :(a

+ &) c + a + & = ac + &c + a + &.P2:o:(a + &)(c +

l) = a(c + l) + &(c + l):o:c + l6[c€]Ts. (2)

(l)(2).o.Theor.

5. a€N.o.lX« = ««

Dem, 1 € [a €] Ts. (1)

a€[a€]Ts.o.lXa = a.o.lXa+l = « + l.D.iX(«
+ l) = a + l.o.a + l€[ae]Ts. (2)

(1) (2) . . Theor.

6. a, & e N . . & « + « = (& + 1) «.

7. a, & € N . . a& = &a. (Eucl. YII, 16)
Dem. a€N.P5.Pl:o.aXl = a = lX«:o:l€[&€]Ts. (1)

a,&eN.&€[&€]Ts:o:a& = &a:D:a&+a=&a+a.Pl.P6
:o:a(& + l) = (& + l)a:D:& + le[&€]Ts. (2)

(l)(2).o.Theor.

8. a, &, c € N . . « (& + c) = a& + ac.
Dem. P 4 . P 7 : . Theor.

9. a, &, c € N . a = & : : «c = &c.

Dem. a, & € N . a = & :: :: 1 € [c e] Ts .-. c e [c e] Ts.o:«c = I)C.a=
& : : ac + a = &c + & : : « (c + 1) = & (c + 1) : : c + 1
6[c€]Ts::o:ccN.o.Ts.

10. a, &, c € N . a < & : . (& — «) c = &c— ac. (Eugl. VII, 7)
Dem. Hyp . o : & — « e N . (& — a) + a = & : o : (& — «) c + ac=bc

: : (& — «) c = &c — ac.

11. a, &, c € N. a < & : : «c < &c.

Dem. Hyp .o:& — aeN.P3:o:(& — «)ceN.P10:o: &c— ac eN
: Thesis.

12. a, &, c e N . .'. <3J < & . = . ac < &c : a = b.=.ac=bc:a > &

. = .ac>bc.

13. a,bya',V el^.a<a' .b<b':o : ab < a' V.

14. a, & e N : . «& . > 'J = . «•

15. a, &,ceN.o.<a^(&c) = a&c.

— 9 —

Dem. a,&€N.Pl:o:l€[c€]Ts. (1)

a, &, c € N . c € [c €j Ts : 3 : a{bc) = abc : o : a {pc^-^-ab = abc

+ ab:o:a{bc-\-b) = ab{c + i):o:a{b{c + i))=ab(c +

l):o:c + le[c€]Ts. (2)

(l)(2).o.Theor.

§ 5. De potestatibus.

Deflnitiones.

1. a€N.o«^*=«.

2. a, & € N . . «^"^* = ^* a.

Theoremaia.

3. a, & € N . . «* € N.

Dem, aeN.Pl:o.l€[ft€]Ts. (1)

a, & € N . & € [& €] Ts : : <3J* € N . § 4 P 3 : : «* « € N . P 1 : :
a6+*€N:o:& + l€[&€jTs. , (2)

(l)(2).o.Theor.

4. a € N . . i" = 1.

5. a, &, c € N . . a^-^<^ = a^ a^

6. a, &, c € N . . {aby = a^ ¥.

7. a, &, c € N . . («*)* = a*^

8. a, &,C€N.o.*.ti < & .m.a^ < ^^ :a = &. = .a^ = &^ :a>&

. = .««>&«.

9. a, &,C€N.a> 1 .o.'.&<c. = .a^ <a^ :&=c.=.a* = a^ :

b>c. = ,a^ ^a'^.

§ 6. De divisione.

Eocplicationes.

Signum / legatur dtoims j)er,

» D » dividit, sive est divisor,
y^ d y> est muUiplex,
» Np » numerus primus,
y^ n » est prtmus cum.

- 10 -

Definitiones,

a, & € N . : & / a = N [it? e] {xa = b).

a, & € N . : a D & . = . & / a - = A-
a, &6N.o:&aa.=i:.aD&.

Np = N [a? e] (9 D 0? . 3 > 1 . 3 < i3? : = a).

a, & € N . :: a Ti & .-. = .-. 3 D a . 3 D & . 3 > 1 : = A-

a, & € N . .-. 3 D (a, &) : = : 9 D a . n . 3 D &.

a, & € N . .-. 3 a («, &) : = : 3 a ^ . ^ • 9 a ^-

a&/c = (a&)/c ; a/ft/c = (a/&)/c ;albXc = (alb) c.

r/jeoremato.

iVoto. Haec theoremata ut in subtractione demonstrantur.

8. a,b,a',b'eN.a = a' .b = b':0'Ci/b = a' Ib',

9. a, &, a', &' 6 N . a = a' . ft = &' : : a D & . = . a' D ^'.

10. a, &, c € N . : ac = & . = . c = & / a.

11. a, & € N . D : a D & . = . & / a € N.

12. a € N . . a / 1 = a.

13. a € N . 3 . a / a = 1.

14. a 6 N . . 1 D a.

15. a € N . . a D a.

16. aybeHi .ab lb = a.

17. a,beJ^.aDb:0'a{bla) = b.

18. a^bjCeN.cDb^^j.aQ) I c) = ab I c.

19. a, &, c € N . a a &c : : a / (bc) = al b I c.

20. aybyCe^^.aa^.bac^o.al (blc) = albXc>

21. a,m,neN .m >*n : q . a'^la'^ = a'^-'^.

22. a^beN.o.aDab.

23. a, ^, c 6 N . a D & . & D c : . a D c.

24. a, &, c € N . a D & D c : . c / a a c / ft.

25. a, &, c € N . c D a . c D & : [j . (a + 2?) / c = a / c 4- & / c.

26. aybyCel:^ .cDa.cDb .a> b:0'{ct-^b) I c^ajc-^b lc.

27. a, &, c € N . c D a . c D & : . c D a + &.

28. a, 6, C€N.cDa.cD&.a>&:o.cDa — ^.

— 11 —

29. a, &, c, m, n e N . c D a . c D & : . c D ma + nb,

30. a, &,c,w, n€N.cDa.cD&.ma>n&:o.cD ma — nb,

31. a, & € N . a D 2> : : a . < w = . 2;.

Z>^. Hyp.Pll.P17.§4P14:o:&/a6N.a(&/a) = &.a<u =
a (6 / a) : . Thesis.

32. a,&€N.aD&.&Da:o.a = &.

33. a€N.o.M9Dai=a.

34. a, & € N . a > & : . 3 D (a, &) = 3 D (&, a — &).

Dem. Hyp. P 28 : o .-. a? D a . o? D & : o : o; D & . a? D (a — &) (1)

Hyp. P 27 : .-. a? D 6 . a? D (a — &) : : iz? D & . a? D (& +(a— V))

: : ^ D & . i3? D a. (2)

(1) (2) : Hyp. o .'. ^ D a . a? D & : =:xm.xT){a—b), (Theor.)

35. a, & € N . : M 3 D (a, &) € N.

Dem, 1 D a . 1 D & : : 3 D (a, &) - = A. (i)

3 D (a, &) . 3 > a : = A. (2)
(l)(2).§3P3:o.Th.

36. a, & € N . . 3 D (a, ft) = 9 D M 9 D (a, &). (Eucl. VII, 2)
Dem. ft = N [c e J (Hp. a < c . & < c : o • Ts.). (1)

a € N . & € N . a < 1 . & < 1 : = A. (2)

(l)(2).0.l€ft. (3)

a, 2^€N.a<c + l.&<c4-l:D.'.^<c.& <c:u:a=c.

b <c:yj:a<c.b = c:^:a = c.h-=zc. (4)

c € fc . a, ft e N . a < c . & < c : : Ts. (5)

ceft.a, &€N.a = c.&<c:o:c€Ar.&<c.a — 2?<c.aD

(a, &) = a D (&, a — &) : : 9 D (&, a — &) = 3 DM9D(&,a— •&)

: : 3 D ( a, &) = 3 D M 3 D (a, & ) : : Ts. (6)

(a, &) [&, a] (6) . c € A . a, & € N . a < c . & = c : : Ts. (7)
ceft.a, &€N.a = c.?^ = c:o:3D(a, 6) = 9Dc = 3DM3Dc

= 9 D M 3 D (a, &) : : Ts. (8)

(4)(5)(6)(7)(8).o.ce/(r.a,&€N.a<c+l.&<c-f-l:o:Ts. (9)
(9)o.C€Ar.o.(c + l)€A-. (10)

(1) (10) . .-. c € N . Hp. a < c . & < c : : Ts. (11)

(a + &) \c\ (11) . : Hp. . Ts. (Theor.)

37. a, &, m € N . . M 9 D {am, bm) = m X M a D (a, 6).

12 —

§ 7. Theoremata varia.

1. a, & € N . a* + &* a 7 : D : a a 7 . & a 7.

2. a? € N . . a? (a? + 1) a 2.

3. o? 6 N . . 07 (o? 4- 1) (^ + 2) a 6.

4. a? € N . D . o? (a? + 1) (2 ir + 1) a 6.

5. 07 € N . : 07 . 7T . 0? + 1.

6. o?€N.d:2o?—1.7t. 207+1.

7. o?€N.o.(2o?+l)*— laS.

8. a € N . a > 1 : .-. Np . 3 > 1 . 3 D a : - = A. (Eucl. VII, 31)

9. a, & € N .-. &* > a .-. 3 D a . 3 > 1 . 3 < & : = A ." D • ^ € Np.

10. a, 6 € N . a€ Np . a - D 6 : : a 7T &. (Edcl. VII, 29)

11. a, &, ceN.aD&c. a7T&:o.aDc.

12. a, & € N . m = M 3 D (a, &) : : a / m . 7T . & / m.

13. a€Np.&, C€N.aD&c:o:aD&.u.aDc. (Eugl. VII, 30)

14. a € Np . &, n € N : : a D &♦» . = . a D 6. (Eugl. IX, 12)

15. a, ^, c € N . a 7T & . c D a : : c 7T &. (Eugl. VII, 23)

16. a, &, C€N.o.*.a7T&.a7Tc: = :a7T&c. (Eugl. VII, 24)

17. a,&, C€N.&7TC.&Da.cDa:o.i5?cDa.

18. a, &, c € N . a 7T & : : 9 D {ac, &) = 3 D (c, &).

19. a, & € N . . K 3 a (<a^» ^) 6 N.

20. a,&€N.o.K3a(«,&) = «2?/M3D(a,&). (Eugl. VII, 34)

21. a, &, c € N . c a a . c a & : : c a K 3 1> («» 2>). (Eugl. VII, 35)

22. o? € N . o? < 41 : . 41 — o? + 07* e Np.

23. M . Np : = A. (EuCL. IX, 20)

24. w € Np . a e N . a - a w • . ^**"^ — 1 a ^- (Fermat)

§ 8. Numerorum rationes.

Eocplicaiiones.
Si p,qe N, tunc - legitur ratio numeri p numero q.

Signum R legitur duorum numerorum ratio, et indicat numeros
ratlonales positivos.

— 13

Definitiones.
1. m,p,^6N.3.w - = mplq,

3. R = ::[a?6]/.i?,^6N.| = a?:- = A-

4. P6N.0. =!=;>.

Theoremata.

5. P,(Z,P',«'6N.o::| = |!. = .pg'=p'«. (Eucl. VII, 19)
i)m. Hp.| = |':o.-.g^«',««'|,g^«'|!6N.P2.-.o.-.g^«'| = ^?'J

.^«' I =V(i''m' ^ =p'q .-. .'.M' =P'^. (1)

Hp.M' =P^Q .'. .*. ^ 6 N . o? - ,0? -,6 N : 0« loopQ^ = xp^q :o

^(«^^««'^(«'l-Vs':^:^^!^^ (2)

(i)(2).a.Th.

6. w, p, ff e N . D . I = ^. (EUCL. VH, 17)

2} t) 1 tn

7. 2}, ^6N.m6N.wD».mDfir:o.— =^^— •

8. P,^,p',«'€N.pTTgr.p'TT^'.|=J:o:p=p'.« = «'.

9. P, «, P', Q'' € N . !>' TT ^' . I = |, : : p' /i> = 3' / ^ = M 3 D (z?, ^).

10. p, q,p\ Q'' 6 N . I =|-, .pnq.q' <q:=is.' (EucL. VH, 21)

11. ??,^,p',^'6N:o:f=f = .f, = |. = .|=f!.(Eu.VII,13)

12. p,^6N.o:: [^€]:m6N.m - 6N.-.- = A.
12^. a^R.o:: [^e] :m6N.ma6N.*.- = a.

— 14 —

13.

13'.
14.

15.

16.
17.

18.

19.
20.

21.
22.
23.

24.

25.
26.

27.
28.

29.

P,«,p',^'€N.D::[(r,5,0€]:r,5,^6N.|=^.J=;- .-. -

a, & € R . : : [(r, 5, ^) e] : r, 5, ^ 6 N . a = y . b= |- .-. - = ^.
a, &, ceR.o:: [{m,n,p,q)^]:m,n,p,q^^,a = - .1) = - . c

p, g, r € N.a= ^ .&= I^ : : a = & . = .1? = (?.

TneN.a, ft6R.a = &. wm e N : o . m& e N.
a, &, c € R . •*. <3J = «.

.-. a = &. = .& = a.

.-. a = & . & = c : . « = c.
NoR.

DeflnUiones.

a,beR.o''^ <b ' = ''-ooel^ .xayXbeN:0'Oca< xb.
a, &€R.o. *&>«. = .«<&.

Theoremata,
p, ^,r6N.a = ^ . & = ^:o:a<&. = .p<gr.

p, g,p', ^' € N.o : I <|? . = .p^' <P'^.

p, (?, r € N .a = ^ . & = ^ : : « < & . = .p > g'.

P..,l>',«'€N.|<|:o.|<|±|<f.

a 6 R . .*. R . 3 > « • - = A.

a € R . .. R . 9 < « .' - = A.

a, & € R . a < 6 : . - R . 3 > « . 3 < & : - = A-

a, & € R : .'. « < ^ . fl^ = ^ * = A.
0.'.a > &.a = &: = A.
0.'.a <^>.a> &: = A-
0.".a-<2?.a-=:&.a->&: = A.

a, &, c € R : .'. ^ < ^ = & . & < c : .' tt < c.
^:.a<b,b <yj = c:^:a<c.

- 15 —

Definitiones,

30. a, & € R . D . a -|- & = [c ej (c € R .*. 0? € N . 0? a, o? &, a?c € N : Oaj.

xa + xb = xc).

31. a, & e R . D : : & — a = .". [o? €] (o? € R . ^i + ^ = *)•

32. a, & € R . . a6 = [c 6] (c € R .'. 0? € N . xa, (wa% a?c € N : o».

(a?a) J) = xc),

33. a, & 6 R . . 2? / a = [o? €] (a? € R . ao? = &).

Theorematd,

34. p,^,reN.D.f+i=^.

35. a, & € R . . a + & 6 R. t

36. P^g^reN.p^gr^o.J-^^^.

37. a, & € R . a < & : . & — a e R.

38. p,«.^',^eN.D.|J = g:.

39. a, & e R . . a& e R.

40. p,«,p',«'€N.o.|/|; = g.

41. a, & € R . . & / a e R.

42. P,g€N.o.|=p/g.

§ 9. Rationalium systemata. Irrationales.

Explicatio,

Si a € K R, signum T a legitur lerminus summus, vel limes sum-
mus ciassis a. Supra boc novum ens relationes ac operationes
tantum deflnimus.

Definitiones.

1. a € K R . o? € R : :: a? < T a . = .-. a . 3 > fl? : - = A-

2. aeKR.ipeR :o:::a? = Ta. = : :a.3 > a?: = a •• weR.w

< ic? : Oi* .-. a . 9 > w : - = A-

3. a€KR.ii?€R:o--^>Ta.=::a?-<Ta.a7- = Ta.

— 16 —

Theorema.

4. ii?eR.D::a? = /.T:R.3 <rr.

Explicatio,
Signum Q legitur qmntitas, numerosque indicat reales positivos,
rationales aut irrationales, et oo exceptis.

Deflnitiones.

5. Q = [xe](aeKR:a- = /^:Rd>Ta.- = /^:T a = x .\

6. a,beQ.o::a = b.=:.\R.d < a: = :R.d <b.

7. a, &€Q.o::a < ^J.m.-.R.a > a.3 <&:- = A.

8. a, &6Q.o:&>a. = .a<&.

Theoremata.

9. a e Q . .-. R . 3 < a : - = A-

10. a e Q . .-. R . 9 > a : - = A-

11. RdQ.

Subsistunt quoque propositiones quae a P 17, 28, 29 in § 8 obti-
nentur, si loco R legatur Q.

Definitiones.

12. a, & e Q . . a + & = T r^ e] {[{x, y) e]:x,y eR.x <a.y <b

.x + y = z.'.- = js).

13. ayb eQ .0 . ab =T [z e]{[{x, y) e\ :XyyeR.x <a.y < b.xy

= z:.- = k)'

Ut valeant hae definitiones, demonstrandum est subsistere pro-
positiones 12 et 13, si a, & e R.

Subtractionem et divisionem ut operationes inversas additionis et
multiplicationis definire licet, illarumque proprietates demonstrare.

§ 10. Quantitatum systemata.

Explicationes.

Si a e K Q, signa I a, E a, L a leguntur: interior, exterior, limes
elassis a.

— 17 —

Definitiones,

1. a€KQ.o.Ia = Q[a?6] ([(w, v) e] :: w, t? e Q .-. t* < o? < v .-.

3 > w . 3 < 1? : : a : • : - = a)-

2. a € K Q . . E a = I (- a).

3. a 6 K Q . . L a = (- 1 a) (- E a).

Theoremata,

4. acKQ.a?, i*,t;€Q.w<a?<i>.(3> w.9<t;:oa):o.a?€la.

5. a€KQ.a?€la:o: [(w> v) €] (w, i? € Q .'. w < o? < i? .-. 9 > w .

3<t;:o:a)- = A-
JDm. P1 = (P4)(P5).

6. a€KQ.t*,V€Q.(3>w.3< t?:o<3j)-'.0.*-3>w.3<^':OI«-
2)^m. P6=:P4.

7. a€KQ.o.I«0^-

8. a€KQ.o.na = Ia.

Lem. Hp.(Ia)[a]P7:o.n«Ol«. (1)

Hp.a?,w,i;€Q.w< a? <t;.(3 > w.a <i>:0^) .P6: o:^,^

€ Q . t* < 0? < t; . (9 > t* . 3 < t; : 1 «)• (2)

Hp. 0? € I a . (2) : : ^ 6 II <3J. (3)

Hp.(3):o:I«on«. (4)

Hp.(l).(4):o:Ts. (Theor.)

9. a, &€KQ.ao&:O.I<3JDl&.

Dem. Hp. a?, t^, i; € Q . w < 0? < t; . (3 > w . 3 < t; : «) : •*. 3 > w .

3 < i; : &. (i)

Hp. a?€la:o:^€l&. (Theor.)

10. a, & € K Q : : 1 {ab) o I a.
Dem. (a&, a) [a, &] P 9 . = . P 10.

11. a,&€KQ.o.I(«&)0(I«)(I&).
Dem. Pll = :P10.n.(&,a)[a,&]P10.

12. a, & € K Q . . 1 « 1 (« ^ &).

13. a,&€KQ.o.IawI&Ol(«^&). .

14. a, & € K Q . . 1 (a&) = (la) (I&).

Dm. Hp. P 11 : . 1 (a&) o ('«) (I&). (1)

Pbano, AriihrMtiC9S principia. 2

18

15.

Dem.

16.

Dem,

17.

Bem.

18.

JDem,

19. •

20.

Dem.

21.

22.

Dem.

23.

24.

Dem,

25.

Dem.

26.

Dem.

26'.

27.
Dem.

Ep.xeQ.u,veQ.u<oo<v.{^>u.^<v:oct) .u\v'eQ
.u' <x<v' .(d>u' .d<v':ob).u''=U(u^Ju') .v" =
IV (^. ^') .' D : ^", t;" € Q . t^" < 0? < I?" .> > u" .^ > v" :o
:al)). (2)

Hp. 0? € la . a? € I& . (2) : 3 . a? € I (a&). (3)

Hp.(3):D:(Ia)(I&)Dl(a&). (4)

Hp.(l).(4):o.Ts.

aeKQ.o.Eao-a.

P15i=(-a)[a]P7.

a 6 K Q . .*. la . Ea : = A-

Hp. P 7 . P 15 : .'. la . Ea : : a - a : = a-

acKQ.o.IEa^Ea.

P17 = (-a)[aJP8.

a, & € K Q . & <3J ' D • Ea ^&.
P18 = (-a,-&)[a,&]P9.
a, & e K Q . : Ea u E& . E (a&).
a, & € K Q . . E (a u &)= (Ea) (F&).
P20 = (-a,-&)[a,&JP14.
aeKQ.o.l^^-a^zrla.
aeKQ.o.'. Ia.La: = A.

.*. E a . L a : = A-

.*. — I <3J . — ^ <3^ . — I-' ^ .' = A-
P22 = P3.

aeKQ.o.ao.I^^La.
a € K Q . . 1 (a L a) = A.

Hp.P14.P7.P22:o:I(aLa).=:.IaILa.o.IaLa. = .A.
a, &€KQ.ao&:D:I'<3J.D.I&^I'&.
Hp. P 18 : : E & E a : : 1 « ^ 1« <3J . D . I & ^ L & : . Ts.
a, & e K Q . : L (ab) o.IaL&ul&LawLaL&.
Hp. : <3^& <a^ . «& & . P 25 : : L (a&) o I a u L a . L (ab) ^lb
u L & : : L (a&) o (I <3j ^ L a) (I & u L &) . L (a&) (I a) (I &) =
L(a&)I(a&)=:A:D:Ts.
a, & e K Q . . L (a&) o L a u L &.
a, &€KQ.o:L(au&)zz:LaE&uL&EauLaL&.
P 27 = (- a, - b) [a, b] P 26.

— 19 —

27' . a, & € K Q . : L (a u &) Q L a u L &.

28. aeKQ.o.LIaoLa.

Dem, Hp. P 7 : : 1 a a . P 25 : : L I a 1 a u L (2. (1)

Hp.P8.P22:o.HaIa = LIaIIa = A. (2)

(l)(2).o.Theor.

28'. a€KQ..o.I'BaO^^«-

29. a € K Q 1 . 1'^^ « r^ I « ^I» E a.

Dem. Hp. : LL a=:L(I a u E a) . P 27' : . Ts.

29'. aeKQ.o.LLaoI-a.

Dem. P 29 . P 28 . P 28' : . Theor.

30. a € K Q . . 1« a = I L a u LL a.

Dem. "R^^.^P^^.o.Jia^lLaOJAia. (1)

Hp.P7:o.ILaoIi<^. (2)

Hp.P29':o.I^l^aoI'a. (3)

(l)(2)(3).o.Theor.

31. aeKQ.o.IilLaoIil^tt.
Dem. P31 = (La)[a]P28.

32. a € K Q . . 1 LL a = A.

Dem. Hp. P 29' : o : I'!' a = L a LL a . (L a) [a] P 24 : o Ts .

33. a€KQ.o:ILILa = A.
Dem. P 31 . P 32 : . P 33.

34. a € K Q . . LLL a = LL a.
Dem. (L a) \a] P 30 . P 32 : o . Theor.

35. a^&eKQ.o.IaL&oI^C^^).
Hp. P 1 4 : . 1 a L & I (oft) = I a I & L & = A. (1)
Hp.P2.P14:o.IaL&E {db) = IaL&I(-au-[;) = I(a

-&)L& = IaE&L2; = A. (2)

(l)(2)oTheor.
a,&eKQ.o.I«I'&^I&LaoI»a&. ( Vide P 26)

Dem.

36

Dem. P36 = :P35.(&,a) [a,&]P35.

37

a,&€KQ.o.EaLfeuE^LaoI'(a^&). (Vide P 27)

Dem. P37 = (-a,-&)[a,??]P36.

38. a, & e K Q . . 1 (a u&) 1 « ^ I & ^ t a L &. ( Vide P 13)

Dem. Hp. ':).l{ayjb)^{la^liiayjl^a)(lhyjlihyjTlh). (1)

Hp.P20.P16:o.I(au&)EaE& = I(au&)E(au^)^A. (2)

— 20 —
Hp. P 37 : : 1 (a u &) (E a L & u E & L a) . . 1 (a u &) L (a u &) .

. = A. (3)

(l)(2)(3).D.Theor.
38'. a, & € K Q . . 15 (a&) E a u E & u L a L &. ( Vide P 19)

39. a € K Q . . 1 L a L I a = A.

Dem. Hp. P 36 : : 1 L a L I a L (L a I a)= A.

40. a € K Q . o^ L I a LL a.
Dem. Hp. P 28 . P 30 . P 39 : o Theor.
40'. aeKQ.o.LEaol-I^^aj.

41. aeKQ.o.I^I-a^I^IawI^Ea.
Bem. P29.P40.P40':o.Theor.

42. aeKQ.o.II^Ia^A.

0.lLEa = A.
O.LLla = LIa.
. LL E a = L E a.

43. a,&€KQ.o.I(I«^I&) = IauI&.

Dem. Hp.P7:o.MI«^I^)Dl«^I&. (1)

Hp.P8.P13:o:I«^I&. = .n«^II&.O.I(I«^I&). (2)
(l)(2)oTheor.

44. a, ft 6 K Q . . 1 (LL a u LL ft) = A.

Dem. Hp. P 38 . P 32 . P 34 : . 1(LL a ^ LL&)oLLaLL&oLLa. (1)
Hp.(l).P8:o.I(tL«^LL&)oIIiLa = A.

45. a€KQ.o.I(Ia^Ka) = IauEa.
Dem. P 8 . P 17 . (- a) [&] P 43 : . Theor.
45'. a€KQ.o.ELa=:IauEa.

46. aeKQ.o.BIa = -(IauLIa).
46'. a € K Q . . BE a = - (E a u L E a).

— 20 —
Hp.P37:o:I(au&)(EaL&uE&La).o.I(au&)L(au&).

= A. (3)

(l)(2)(3).o.Theor.
38'. a,&€KQ.o.l?(a&)DEauE&uLaL&. ( Vide P 19)

39. a € K Q . . 1 1* a I» I « = A.

Dem. Hp. P 36 : : 1 1» « 1« I « D I*(J^ « I «)= A.

40. a 6 K Q . o^ 1« I « 1»!» «.
Dem. Hp.P28.P30.P39:oTheor.

40'. aeKQ.o.I^EaoI''^^. '

41. a€KQ.o.l«tia = LlauLEa. |
Dem. P29.P40.P40':o.Theor. ,

42. aeKQ.o.II^I^^A.

O.ILEa^A. I

O.LLla = LIa. I

. LL E a = L E a.

43. a,&€KQ.o.l(I«^l&) = I«uI&.

Dem, Hp.P7:o.I(Ia^I&)Dl«^I&. (1)

Hp.P8.P13:o:Ia^I^. = .IIa^II&.D.I(I«^I&). (2) j
(l)(2)oTheor.

44. a,&€KQ.o.I(LLauLL6) = A. i
Dm. Hp.P^^.P^^.P^^^o.I^LLauLL&^oLLaLL&oI'!'^. (1) l

Hp.^l^.P^^o.I^I-I^a^I^Ii&^OlLLa^A. I

45. a€KQ.o.I(IauEa)==IauEa. j
Dem. P8.P17. (-a)[&]P43:o.Theor. i
45'. a€KQ.o.ELa = IauEa. '

a€KQ.o.BI« = -(IauLIa). '

a€KQ.o.BEa = -(EauLEa).

46.
46'.

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